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Structural instabilities and transverse effective charges in IV–VI compounds
Solid State Communications, Vol. 21, pp. 521-524,
Pergamon Press.
1977.
Printed in Great Britain
STRUCTURAL INSTABILITIES AND TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS S. Katayama and H. Kawamura Department of Physics, Osaka University, Toyonaka, 560 Japan (Received 14 October 1976 by Y. Toyozawa) Transverse effective charges for some IV-VI, III-V and II-VI compounds are investigated on the basis of the interband electron-phonon coupling with the use of the Penn model for electron energy bands. The results show that the large transverse effective charges of IV-VI compounds are related with strong electron-phonon interaction as well as with the softening of TO phonon in IV-VI compounds. THE STRUCTURAL instabilities and the phase transitions from NaCl-structure to arsenic structure in IV-VI compounds are closely related with the softening of the TO phonon at the zone center. In the average V compounds, the valence ns2electrons are well separated from &electrons, so that only three npelectrons contribute to the bonding.’ On the other hand, there are six nearest neighbours in the NaCl-structure. L&~ovskyet al. 2 described this situation in terms of the resonating bond. Using this model, they explained the large transverse effective charges in IV-VI compounds. The softening of TO phonon can also be understood as the weakening of the restoring force resulting from the reorientation of the resonating bond synchronized with the lattice vibration. From the band-view point, the phase transition through the softening of TO phonon was explained in terms of the strong interband electron-TO phonon coupling. 3*4 In this letter, we shall derive an expression for the transverse effective charge from the band picture, and present a microscopic picture for the structural instability of IV-VI compounds. We can describe the phase transition of IV-VI compounds by the Hamiltonian composed of the unperturbed energies of the electrons and phonons together with the interband electron-TO phonon interaction and the fourth order phonon-phonon coupling which leads to the anharmonic oscillation of the lattice; H
=
c ‘%k~;k~nk n,k
+ ’ c 2 a,i
(%&,.i
+
&Q&Q,,&
k’,k CL
where c& c&, Q, and Pd are the creation and annihilation operators for electrons and the normal coordinate of phonons and its conjugate momentum, respectively.
The band index of the Bloch electrons is specified with n. The vectors k and q are the wave vectors of electrons and phonons. M, N and a are the reduced mass of the ion-pair, the number of unit cells and lattice constant, respectively. The interband optical deformation potential is denoted by5. c?is the unit polarization vector for TO phonon. As pointed out in a previous note,* the TO phonon frequency at the zone center is reduced by the renormalization of the interband electron-TO phonon coupling. This coupling also induces the interband polarization of valence electrons. For the calculation, we utilize the Bogoliubov transformation method. This gives rise to a new electronic state @ in the mean field approximation which is expressed by the linear combination of the normal Bloch states. The expectation value of c&,k is obtained as
where ( ) means the statistical average over the electronic states a, and the renormalized electronic energies are
(3) In deriving the above formula we confined ourselves to the two bands [conduction (c) and valence (v) bands] and the TO phonons at the zone center. The function f(E) is the Fermi distribution function. The expectation value ( Qe ) represents a static displacement of the sublattice along (11 l)-direction. It becomes finite below the transition temperature, inducing a static change of the valence electron density ( c&,k >c through equation (2). We can obtain the expectation values ( Qe ) and 521
522
TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS
Vol. 21, No. 6
Table 1. Transverss effective charge e& optical dielectric constant e_, average energy gap Eo, average optical deformation potential X. TOphonon f;equency Q,, mechanical jiequency wo. reduction of TO phonon frequency due to electron-phonon coup&y Aw , and localized effective charge q , for IV- VZ,ZZZ-V and ZZ- VI compounds. The unit for a%, w. and Ao2 is e’/Mr& where r. is the atomic distance for nearest neighbours e2e
em
EG
(ev)
f (eV)
G?j.
wa
Au2
G/e Present
Lucovsky
1.12
2.16
1.54 1.54 1.64
2.40 2.3 1 2.29
SnTe (1OOK) PbS PbSe PbTe (1.6K)
8.1 9.1 4.8 5.8 6.5 7.2
45.0 54.0 17.2 22.9 32.8 39.0
2.33 2.03 4.03 3.31 2.54 2.32
17.5 17.4 16.2 16.8 14.8 15.2
0.33 -0 0.78 0.81 0.70 0.22
AlAs AlSb GaP GaAs InP InAs InSb
2.3 1.9 2.0 2.2 2.6 2.5 2.5
9.0 10.2 8.5 10.9 9.6 12.3 15.7
5.2 4.7 5.6 5.2 5.2 4.6 3.7
8.3 5.4 7.2 7.4 9.9 8.6 6.8
11.7 10.9 9.8 9.8 9.5 10.1 10.9
13.8 12.3 11.7 11.3 13.4 12.2 12.0
1.3 0.6 0.9 0.9 1.8 1.3 0.9
0.60 0.61 0.70 0.45 1.oo 0.63 0.38
0.81 0.96 0.78 1.36 1.01 0.53
ZnS ZnSe ZnTe
2.2 2.0 2.0
5.1 5.9 7.3
7.8 7.1 5.8
1.4 0.3 0
5.4 3.2 6.4
11.8 8.3 11.6
0.034 0.001 0
1.25 1.55 1.58
1.75 1.56 1.58
(c&c,~)~ self-consistently by combining equation (2) with the static part of an equation of motion for Q. with the Hamiltonian given by equation (1). Thus we can get the behaviors of the order parameters (Q, 1as well as (c+cjo below the transition temperature. The details of the calculation will be published elsewhere. In this paper we are interested in the valence charge redistribution induced by optic phonon vibration. Suppose the sublattices of NaCl-structure are displaced by a small amount u in the opposite directions to each other along x-axis. The transverse effective charge due to the redistribution of the valence electrons is defined by
10.8 11.4 12.4 12.3
7.9 7.8 5.6 6.6 6.0 6.3
where \knk(r) is the Bloch function. The expectation value (&c& >or the change of electron density p(t, u) is proportional to the lattice displacement u = (l/a)Q,& as shown in equation (2), where 2%is the x-component of the unit polarization vector for TO phonon. Putting equations (6) and (5) into equation (4) we eet
We calculated equation (7) with an use of the Penn model,5 in which the energy gap is assumed to be at the spherical Fermi surface which contains the whole valence electrons. We obtain
(4) where P”(u) is the induced electronic dipole per unit cell in the direction of the displacement. It is given by P’(u) = -f
1 x&r, u) dr,
where x is the coordinate along the displacement, and p(r, u) is the change of the electron density given by
(5)
adding the contribution from the rigid ion core: e(Z, - Z2)/2 where Zi and Z2 are the valences of two ions at the lattice sites. Here kF, Eo and n are the Fermi wave number, the average energy gap in Penn model and the number of valence electrons per unit cell, respectively. The average value of the deformation potential over the Fermi surface is denoted as f. If the relation
Vol.21,No.6
TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS
is used for EG in equation (8), where e_,is the optical dielectric constant and op = dm is the plasma frequency of valence electrons, we get a linear relation between e; -e(Z, -ZZ,)/2anddaaswasshown by Lucovsky et al.2 8 The transverse effective charge is related with LOand TO-phonon frequencies as wi - Gg = 4nNeG*/ME,. The experimental values of ec estimated from this relation, the average energy gap obtained from equation (9) and the deformation potential calculated from equation (8) are shown in Table 1 for some IV-VI compounds together with some III-V and II-VI compounds. Almost all the experimental data at room temperature were taken from references 6 and 7. For Gj, of PbTe at 1.6 K, we employed our recent data,a extrapolating to zero carrier density. For SnTe we assumed that GjT is zero at the transition temperature which is about 100 K.’ For IV-VI compounds, the TO phonon frequency is sensitive to temperature. However, in the above calculation, the effects of temperature are neglected. It should be noted from the table that the large transverse effective charge for IV-VI compounds are related with the large interband deformation potential as well as with the small average band gap. The optical deformation potential just obtained, can be used for the calculation of the frequency of the soft TO phonon. Substituting equation (2) into the interaction Hamiltonian in equation (1) we get the renormalized TO phonon frequency4
The atomic displacements of optical type couple to each other through the polarizations of valence electrons. This gives rise to the reduction of TO-phonon frequency as expressed by - Aw2 . There will be some localized charges on the lattice sites as in the case of the ionic crystals, where the classical dipole field corrections are valid for the TO phonon frequency. Lucovsky et al6 have shown that the TO phonon frequency is given by 0% = w$ -a&,.
(11)
Here o. is the mechanical frequency which is related to characterizes the reduction
the compressibility, and ano
523
of the restoring force due to dipole-dipole interaction. The latter is given in terms of a localized effective charge e: as s2& = 4nNe:* /3M. Putting the expression for the TO phonon frequency in equation (11) into the unperturbed frequency in equation (lo), we re-examined the values of the localized effective charge. For o. we used the values obtained by Lucovsky et al.’ For the computation of Aa’, we employed the Penn model which yields Aw2 = (n/Ma2)@2/2EF). From the difference between the observed value of S& and the calculated value of c& - Aa’, we obtained the localized effective charge e: as shown in the table. The present values of localized effective charge are much smaller than those obtained by Lucovsky et al. ,6 which are shown in the last column. They did not take into account the effect of the interband electron-TO phonon interaction which is quite essential to reduce the TO phonon frequency as explored in this letter. According to Burstein and others,‘*lovll, the effective charge due to the valence electrons in semiconductor is not effective to produce the classical Lorentz field. So, it is reasonable that the present values of localized effective charge are of the order of one for IV-VI and III-V compounds and less than two for II-VI compounds, although the transverse effective charge which includes the effect of valence electrons is quite large especially for IV-VI compounds. The most remarkable feature of the present result is that the values of Aw2 which is responsible for the softening of TO phonon, leading to the lattice instability, are quite large for IV-VI compounds, while they are very small for III-V and II-VI compounds. The above discussion is valid only at low temperature where the effect of lattice anharmonicity and the energy distribution of valence electrons are neglected. However, we were obliged to use the room temperature data, since the low temperature data available are very few. Our discussion was confined to the stoichiometric materials where the free carriers are absent. The presence of the free carriers decreases the transverse effective charge and weakens the effect of TO phonon softening by reducing the effective number of the transitions from the valence band to the conduction band.’ Acknowledgement - We are indebted to Dr. K. Murase
for the stimulating discussions.
REFERENCES 1.
COHEN M.H., FALICOV L.M. & GOLIN S., IBMJ. Res. Dev. 8,215 (1964).
2.
LUCOVSKY G. 8z WHITE R.M., P&s. Rev. B8,660 (1973).
3.
KRISTOFFEL N. & KONSIN P., Phys. Status Solidi 28,73 1 (1968).
4.
KAWAMURA H., KATAYAMA S., TAKANO S. & HOTTA S., Solid Stare Commun. 14,259 (1974).
524
TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS
Vol. 21, No. 6
5.
PENN D.R.,Phys. Rev. 128,2093 (1962).
6.
LUCOVSKY G., MARTIN R.M. & BURSTEIN E.,Phys. Rev. J&I,1367 (1971).
7.
BURSTEIN E., PINCZUK A. & WALLIS R.F., hoc. Con& Phys. of SemimetalsandNarrow GapSemiconductor (Edited by CARTER D.L. & BATE R.T.), p. 251. Pergamon Press, New York (1971).
8.
KAWAMURA H., NISHIKAWA S. & NISHI S., Proc. 13th Int. Con&Phys. Semicond. (to be published).
9.
IIZUMI M., HAMAGUCHI Y., KOMATSUBARA K.F. & KATO Y.,J. Phys. Sot. Japan 38,443 (1975).
10.
STERN F., in Solid State Physics(Edited by SEITZ F. & TURNBULL D.), Vol. 15, p. 343. Academic Press, New York.
11.
BRODSKY M.H. & BURSTEIN E,, J. Phys. Chem. Solids 28,1655 (1967).
Pergamon Press.
1977.
Printed in Great Britain
STRUCTURAL INSTABILITIES AND TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS S. Katayama and H. Kawamura Department of Physics, Osaka University, Toyonaka, 560 Japan (Received 14 October 1976 by Y. Toyozawa) Transverse effective charges for some IV-VI, III-V and II-VI compounds are investigated on the basis of the interband electron-phonon coupling with the use of the Penn model for electron energy bands. The results show that the large transverse effective charges of IV-VI compounds are related with strong electron-phonon interaction as well as with the softening of TO phonon in IV-VI compounds. THE STRUCTURAL instabilities and the phase transitions from NaCl-structure to arsenic structure in IV-VI compounds are closely related with the softening of the TO phonon at the zone center. In the average V compounds, the valence ns2electrons are well separated from &electrons, so that only three npelectrons contribute to the bonding.’ On the other hand, there are six nearest neighbours in the NaCl-structure. L&~ovskyet al. 2 described this situation in terms of the resonating bond. Using this model, they explained the large transverse effective charges in IV-VI compounds. The softening of TO phonon can also be understood as the weakening of the restoring force resulting from the reorientation of the resonating bond synchronized with the lattice vibration. From the band-view point, the phase transition through the softening of TO phonon was explained in terms of the strong interband electron-TO phonon coupling. 3*4 In this letter, we shall derive an expression for the transverse effective charge from the band picture, and present a microscopic picture for the structural instability of IV-VI compounds. We can describe the phase transition of IV-VI compounds by the Hamiltonian composed of the unperturbed energies of the electrons and phonons together with the interband electron-TO phonon interaction and the fourth order phonon-phonon coupling which leads to the anharmonic oscillation of the lattice; H
=
c ‘%k~;k~nk n,k
+ ’ c 2 a,i
(%&,.i
+
&Q&Q,,&
k’,k CL
where c& c&, Q, and Pd are the creation and annihilation operators for electrons and the normal coordinate of phonons and its conjugate momentum, respectively.
The band index of the Bloch electrons is specified with n. The vectors k and q are the wave vectors of electrons and phonons. M, N and a are the reduced mass of the ion-pair, the number of unit cells and lattice constant, respectively. The interband optical deformation potential is denoted by5. c?is the unit polarization vector for TO phonon. As pointed out in a previous note,* the TO phonon frequency at the zone center is reduced by the renormalization of the interband electron-TO phonon coupling. This coupling also induces the interband polarization of valence electrons. For the calculation, we utilize the Bogoliubov transformation method. This gives rise to a new electronic state @ in the mean field approximation which is expressed by the linear combination of the normal Bloch states. The expectation value of c&,k is obtained as
where ( ) means the statistical average over the electronic states a, and the renormalized electronic energies are
(3) In deriving the above formula we confined ourselves to the two bands [conduction (c) and valence (v) bands] and the TO phonons at the zone center. The function f(E) is the Fermi distribution function. The expectation value ( Qe ) represents a static displacement of the sublattice along (11 l)-direction. It becomes finite below the transition temperature, inducing a static change of the valence electron density ( c&,k >c through equation (2). We can obtain the expectation values ( Qe ) and 521
522
TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS
Vol. 21, No. 6
Table 1. Transverss effective charge e& optical dielectric constant e_, average energy gap Eo, average optical deformation potential X. TOphonon f;equency Q,, mechanical jiequency wo. reduction of TO phonon frequency due to electron-phonon coup&y Aw , and localized effective charge q , for IV- VZ,ZZZ-V and ZZ- VI compounds. The unit for a%, w. and Ao2 is e’/Mr& where r. is the atomic distance for nearest neighbours e2e
em
EG
(ev)
f (eV)
G?j.
wa
Au2
G/e Present
Lucovsky
1.12
2.16
1.54 1.54 1.64
2.40 2.3 1 2.29
SnTe (1OOK) PbS PbSe PbTe (1.6K)
8.1 9.1 4.8 5.8 6.5 7.2
45.0 54.0 17.2 22.9 32.8 39.0
2.33 2.03 4.03 3.31 2.54 2.32
17.5 17.4 16.2 16.8 14.8 15.2
0.33 -0 0.78 0.81 0.70 0.22
AlAs AlSb GaP GaAs InP InAs InSb
2.3 1.9 2.0 2.2 2.6 2.5 2.5
9.0 10.2 8.5 10.9 9.6 12.3 15.7
5.2 4.7 5.6 5.2 5.2 4.6 3.7
8.3 5.4 7.2 7.4 9.9 8.6 6.8
11.7 10.9 9.8 9.8 9.5 10.1 10.9
13.8 12.3 11.7 11.3 13.4 12.2 12.0
1.3 0.6 0.9 0.9 1.8 1.3 0.9
0.60 0.61 0.70 0.45 1.oo 0.63 0.38
0.81 0.96 0.78 1.36 1.01 0.53
ZnS ZnSe ZnTe
2.2 2.0 2.0
5.1 5.9 7.3
7.8 7.1 5.8
1.4 0.3 0
5.4 3.2 6.4
11.8 8.3 11.6
0.034 0.001 0
1.25 1.55 1.58
1.75 1.56 1.58
(c&c,~)~ self-consistently by combining equation (2) with the static part of an equation of motion for Q. with the Hamiltonian given by equation (1). Thus we can get the behaviors of the order parameters (Q, 1as well as (c+cjo below the transition temperature. The details of the calculation will be published elsewhere. In this paper we are interested in the valence charge redistribution induced by optic phonon vibration. Suppose the sublattices of NaCl-structure are displaced by a small amount u in the opposite directions to each other along x-axis. The transverse effective charge due to the redistribution of the valence electrons is defined by
10.8 11.4 12.4 12.3
7.9 7.8 5.6 6.6 6.0 6.3
where \knk(r) is the Bloch function. The expectation value (&c& >or the change of electron density p(t, u) is proportional to the lattice displacement u = (l/a)Q,& as shown in equation (2), where 2%is the x-component of the unit polarization vector for TO phonon. Putting equations (6) and (5) into equation (4) we eet
We calculated equation (7) with an use of the Penn model,5 in which the energy gap is assumed to be at the spherical Fermi surface which contains the whole valence electrons. We obtain
(4) where P”(u) is the induced electronic dipole per unit cell in the direction of the displacement. It is given by P’(u) = -f
1 x&r, u) dr,
where x is the coordinate along the displacement, and p(r, u) is the change of the electron density given by
(5)
adding the contribution from the rigid ion core: e(Z, - Z2)/2 where Zi and Z2 are the valences of two ions at the lattice sites. Here kF, Eo and n are the Fermi wave number, the average energy gap in Penn model and the number of valence electrons per unit cell, respectively. The average value of the deformation potential over the Fermi surface is denoted as f. If the relation
Vol.21,No.6
TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS
is used for EG in equation (8), where e_,is the optical dielectric constant and op = dm is the plasma frequency of valence electrons, we get a linear relation between e; -e(Z, -ZZ,)/2anddaaswasshown by Lucovsky et al.2 8 The transverse effective charge is related with LOand TO-phonon frequencies as wi - Gg = 4nNeG*/ME,. The experimental values of ec estimated from this relation, the average energy gap obtained from equation (9) and the deformation potential calculated from equation (8) are shown in Table 1 for some IV-VI compounds together with some III-V and II-VI compounds. Almost all the experimental data at room temperature were taken from references 6 and 7. For Gj, of PbTe at 1.6 K, we employed our recent data,a extrapolating to zero carrier density. For SnTe we assumed that GjT is zero at the transition temperature which is about 100 K.’ For IV-VI compounds, the TO phonon frequency is sensitive to temperature. However, in the above calculation, the effects of temperature are neglected. It should be noted from the table that the large transverse effective charge for IV-VI compounds are related with the large interband deformation potential as well as with the small average band gap. The optical deformation potential just obtained, can be used for the calculation of the frequency of the soft TO phonon. Substituting equation (2) into the interaction Hamiltonian in equation (1) we get the renormalized TO phonon frequency4
The atomic displacements of optical type couple to each other through the polarizations of valence electrons. This gives rise to the reduction of TO-phonon frequency as expressed by - Aw2 . There will be some localized charges on the lattice sites as in the case of the ionic crystals, where the classical dipole field corrections are valid for the TO phonon frequency. Lucovsky et al6 have shown that the TO phonon frequency is given by 0% = w$ -a&,.
(11)
Here o. is the mechanical frequency which is related to characterizes the reduction
the compressibility, and ano
523
of the restoring force due to dipole-dipole interaction. The latter is given in terms of a localized effective charge e: as s2& = 4nNe:* /3M. Putting the expression for the TO phonon frequency in equation (11) into the unperturbed frequency in equation (lo), we re-examined the values of the localized effective charge. For o. we used the values obtained by Lucovsky et al.’ For the computation of Aa’, we employed the Penn model which yields Aw2 = (n/Ma2)@2/2EF). From the difference between the observed value of S& and the calculated value of c& - Aa’, we obtained the localized effective charge e: as shown in the table. The present values of localized effective charge are much smaller than those obtained by Lucovsky et al. ,6 which are shown in the last column. They did not take into account the effect of the interband electron-TO phonon interaction which is quite essential to reduce the TO phonon frequency as explored in this letter. According to Burstein and others,‘*lovll, the effective charge due to the valence electrons in semiconductor is not effective to produce the classical Lorentz field. So, it is reasonable that the present values of localized effective charge are of the order of one for IV-VI and III-V compounds and less than two for II-VI compounds, although the transverse effective charge which includes the effect of valence electrons is quite large especially for IV-VI compounds. The most remarkable feature of the present result is that the values of Aw2 which is responsible for the softening of TO phonon, leading to the lattice instability, are quite large for IV-VI compounds, while they are very small for III-V and II-VI compounds. The above discussion is valid only at low temperature where the effect of lattice anharmonicity and the energy distribution of valence electrons are neglected. However, we were obliged to use the room temperature data, since the low temperature data available are very few. Our discussion was confined to the stoichiometric materials where the free carriers are absent. The presence of the free carriers decreases the transverse effective charge and weakens the effect of TO phonon softening by reducing the effective number of the transitions from the valence band to the conduction band.’ Acknowledgement - We are indebted to Dr. K. Murase
for the stimulating discussions.
REFERENCES 1.
COHEN M.H., FALICOV L.M. & GOLIN S., IBMJ. Res. Dev. 8,215 (1964).
2.
LUCOVSKY G. 8z WHITE R.M., P&s. Rev. B8,660 (1973).
3.
KRISTOFFEL N. & KONSIN P., Phys. Status Solidi 28,73 1 (1968).
4.
KAWAMURA H., KATAYAMA S., TAKANO S. & HOTTA S., Solid Stare Commun. 14,259 (1974).
524
TRANSVERSE EFFECTIVE CHARGES IN IV-VI COMPOUNDS
Vol. 21, No. 6
5.
PENN D.R.,Phys. Rev. 128,2093 (1962).
6.
LUCOVSKY G., MARTIN R.M. & BURSTEIN E.,Phys. Rev. J&I,1367 (1971).
7.
BURSTEIN E., PINCZUK A. & WALLIS R.F., hoc. Con& Phys. of SemimetalsandNarrow GapSemiconductor (Edited by CARTER D.L. & BATE R.T.), p. 251. Pergamon Press, New York (1971).
8.
KAWAMURA H., NISHIKAWA S. & NISHI S., Proc. 13th Int. Con&Phys. Semicond. (to be published).
9.
IIZUMI M., HAMAGUCHI Y., KOMATSUBARA K.F. & KATO Y.,J. Phys. Sot. Japan 38,443 (1975).
10.
STERN F., in Solid State Physics(Edited by SEITZ F. & TURNBULL D.), Vol. 15, p. 343. Academic Press, New York.
11.
BRODSKY M.H. & BURSTEIN E,, J. Phys. Chem. Solids 28,1655 (1967).